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Math  /  Algebra

QuestionThe height, hh, in centimetres, of a piston moving up and down in an engine cylinder can be modelled by the function h(t)=18sin⁡(50πt)+18h(t)=18 \sin (50 \pi t)+18, where tt is the time, in seconds. What is the piston's minimum height? a) -18 cm b) 9 cm c) 0 cm d) 18 cm

Studdy Solution

STEP 1

1. The function h(t)=18sin⁡(50πt)+18 h(t) = 18 \sin(50 \pi t) + 18 models the height of the piston.
2. The sine function, sin⁡(x)\sin(x), oscillates between -1 and 1.
3. We need to find the minimum value of the function h(t) h(t) .

STEP 2

1. Analyze the sine function's range.
2. Determine the minimum value of h(t) h(t) based on the sine function's minimum.
3. Identify the piston's minimum height from the given options.

STEP 3

The sine function, sin⁡(x)\sin(x), has a range of [−1,1][-1, 1]. This means that the value of sin⁡(50πt)\sin(50 \pi t) will also oscillate between -1 and 1.

STEP 4

To find the minimum value of h(t)=18sin⁡(50πt)+18 h(t) = 18 \sin(50 \pi t) + 18 , we need to consider the minimum value of sin⁡(50πt)\sin(50 \pi t), which is -1.
Substitute sin⁡(50πt)=−1\sin(50 \pi t) = -1 into the function:
h(t)=18(−1)+18=−18+18=0 h(t) = 18(-1) + 18 = -18 + 18 = 0

STEP 5

The piston's minimum height is the minimum value of h(t) h(t) , which we calculated to be 0 cm.
Therefore, the correct answer is:
c) 0 cm
The piston's minimum height is 0 \boxed{0} cm.

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